Re: Non-arbitrary mathematics

On 10/24/2006 4:40 PM, Frederick Williams wrote:
Eckard Blumschein wrote:

On 10/19/2006 3:32 PM, Eckard Blumschein wrote:

Admittedly, I consider the common credo of Dedekind and G. Cantor "there
are more real than rational numbers" wrong, I vote for oo+a=oo not for
oo+a>oo, and I do so in order to avoid arbitrary nonsense.

Since perhaps nobody can imagine how to make mathematics non-arbitrary,
I will tell just one example:

Mathematicians told me how to deal with the nil between IR+ and IR- if
IR is cut into these two parts. Every gave his own recommendation.
Nobody was able to convincingly explain why his just suggestion was the
correct one. They all agreed on beeing allowed to decide arbitrarily.

What do you take correctness to be?

One starts with axioms and deduces theorems.

Sorry, I am not a mathematician. I am a user of mathematics who does not
like sloppy theories. Forget those axioms which were only fabricated
e.g. by Zermelo in order to defend an essentially utopic set theory.

You may call the axioms
"arbitrary" but if they are sound

Is the axiom of extensionality really sound?
Has the reformulation of the Archimedes axiom like an assertion of
existence really improved it?

and the rules of inference preserve
soundness and if (this is most important) the theorems are interesting,
then that is some justification for them.

Completeness is a bonus unobtainable, it would seem, in the interesting
cases.

I do not understand what you intend to express here.
After sharing a widespread lack of understanding, I found out that the
rational numbers cannt be completed at all. They are ideally perform
what numbers can do. In this sense they are complete. Of course,
incommensurable relations evade any numerical representation.

I only guess, but notice I very rarely went wrong with guesses of such
kind, analysis can use what I call real "numbers" as if they were really
numbers. Each of them is a fictitious solution to a task tha t cannot be
fulfilled at all. This is true not just for irrational "numbers" but
also for any fictitious numerical representation with an actually
infinite amount of nuimbers, no matter what system of numbers you
prefer.


Applicability to the real world is surprising (to me) but not
infrequent.

Mathematical basics obviously fit best to physics if they fit to
mathematics as a whole.

One thinks of Riemannian geometry and general relativity;
matrices and quantum theory

What about quantum mechanics, I blame Schroedinger and Weyl for
misinterpretation of a solution that has to include seemingly unphysical
und redundant components within complex plane in order to code the
correct one-sidedness in the real world.

I don't know if the theory of transfinite numbers (Cantor's in essence
but rigorously axiomatized)

This rigorosity reminds me of rigor mortis. It has been nothing but an
overly promoted slick substitute for Cantor´s abandoned definition of a
set, implying the mutually excluding potential and actual infinity at a
time.

has applications in physics. Maybe it
doesn't but just maybe it will.

The only good thing of set theory might be: It is nearly harmless nonsense.

I seem to recall Karl Popper
speculating that Kant's antinomies of infinity might be solved using
Robinson's nonstandard numbers.

So far, Robinson's *R has perhaps not furnished any evidence for
providing something valuable. Admittedly, I did not expect them valuable
because in my unterstandig, infinity cannot be topped.

I regard Hegel's antinomies still valuable requests for deeper thoughts.
Are you ready to develop and suggest own ideas? Or might you be willing
to check those of mine? I would be delighted.

Yours sincerely,
Eckard Blumschein



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